On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids

Krischok, Andreas; Linder, Christian

doi:10.1002/nme.5120

Cited by 31 publications

(39 citation statements)

References 42 publications

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“…Remark The center evaluation is, eg, used in . A possible alternative is the use of the average deformation gradient

{boldF}_{a v g} = \frac{1}{V} \int_{{normalΩ}_{e}} {boldF}^{h, e} normald V

within an element with volume V as eg done in, the works . However, differences between using the average value and the evaluation at the element centroid are very small in usual problems without localization of strains (see Section 3.7).…”

Section: Enhanced Assumed Strain Methodsmentioning

confidence: 99%

“…In general, they improve the bending behavior in initially distorted meshes. One well‐working and widely used possibility (cf, eg, works) is

{boldM}_{I} false(boldX false) = \frac{j_{0}}{j^{h, e} false(𝛏 false)} {boldJ}_{0}^{- normalT} {\overset{boldM}{true}}_{I} false(𝛏 false) {boldJ}_{0}^{- 1},

where j h , e , j 0 and J 0 are given in and , respectively. The special structure with the elementwise constant quantities j 0 and J 0 ensures that the patch test is fulfilled by construction of the enhanced field (see Section 2.4).…”

Section: Enhanced Assumed Strain Methodsmentioning

confidence: 99%

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On transformations and shape functions for enhanced assumed strain elements

Pfefferkorn

Betsch

2019

Numerical Meth Engineering

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Summary We summarize several previously published geometrically nonlinear EAS elements and compare their behavior. Various transformations for the compatible and enhanced deformation gradient are examined. Their effect on the patch test is one main concern of the work, and it is shown numerically and with a novel analytic proof that the improved EAS element proposed by Simo et al in 1993 does not fulfill the patch test. We propose a modification to overcome that drawback without losing the favorable locking‐free behavior of that element. Furthermore, a new transformation for the enhanced field is proposed and motivated in a curvilinear coordinate frame. It is shown in numerical tests that this novel approach outperforms all previously introduced transformations.

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“…Remark The center evaluation is, eg, used in . A possible alternative is the use of the average deformation gradient

{boldF}_{a v g} = \frac{1}{V} \int_{{normalΩ}_{e}} {boldF}^{h, e} normald V

Section: Enhanced Assumed Strain Methodsmentioning

confidence: 99%

“…In general, they improve the bending behavior in initially distorted meshes. One well‐working and widely used possibility (cf, eg, works) is

{boldM}_{I} false(boldX false) = \frac{j_{0}}{j^{h, e} false(𝛏 false)} {boldJ}_{0}^{- normalT} {\overset{boldM}{true}}_{I} false(𝛏 false) {boldJ}_{0}^{- 1},

Section: Enhanced Assumed Strain Methodsmentioning

confidence: 99%

On transformations and shape functions for enhanced assumed strain elements

Pfefferkorn

Betsch

2019

Numerical Meth Engineering

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“…The key step to simplify the analysis is to take advantage of the fact that the spaces

{bold-italicV}_{bold-italicu}^{h}

{bold-italicV}_{p}^{h}

, and V λ are all finite dimensional and spanned by the basis function (cf. ). This fact allows us to construct a bijective map from the finite dimensional spaces

{bold-italicV}_{bold-italicu}^{h}

{bold-italicV}_{p}^{h}

, and V λ onto the Euclidean space of the nodal solution.…”

Section: Numerical Stabilitymentioning

confidence: 97%

“…In poromechanics, it is well known that the inf–sup condition arises when the pore fluid imposes an incompressibility constraint in the solid deformation—which is common especially in the early stages of loading. As such, mixed finite elements that employ equal‐order interpolations for the displacement and pore pressure fields may result in spurious oscillations in the pore pressure field, see for example.…”

Section: Numerical Stabilitymentioning

confidence: 99%

Mixed Arlequin method for multiscale poromechanics problems

Sun

Cai

Choo

2017

Numerical Meth Engineering

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Summary An Arlequin poromechanics model is introduced to simulate the hydro‐mechanical coupling effects of fluid‐infiltrated porous media across different spatial scales within a concurrent computational framework. A two‐field poromechanics problem is first recast as the twofold saddle point of an incremental energy functional. We then introduce Lagrange multipliers and compatibility energy functionals to enforce the weak compatibility of hydro‐mechanical responses in the overlapped domain. To examine the numerical stability of this hydro‐mechanical Arlequin model, we derive a necessary condition for stability, the twofold inf–sup condition for multi‐field problems, and establish a modified inf–sup test formulated in the product space of the solution field. We verify the implementation of the Arlequin poromechanics model through benchmark problems covering the entire range of drainage conditions. Through these numerical examples, we demonstrate the performance, robustness, and numerical stability of the Arlequin poromechanics model. Copyright © 2016 John Wiley & Sons, Ltd.

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“…For instance, for ν = 0.49 even 12 quadratic elements (142 DOFs) are more accurate than 48000 linear elements (107998 DOFs). It is well-known that in the incompressibility limit ν → 0.5 classical quadratic elements show locking behavior and mixed, enhanced, or stabilized formulations [23,33,44,[50][51][52]60,62] shall be used.…”

Section: Numerical Investigation By Eigenvalue Analysismentioning

confidence: 99%

Computational aspects of growth-induced instabilities through eigenvalue analysis

et al. 2015

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The objective of this contribution is to establish a computational framework to study growth-induced instabilities. The common approach towards growth-induced instabilities is to decompose the deformation multiplicatively into its growth and elastic part. Recently, this concept has been employed in computations of growing continua and has proven to be extremely useful to better understand the material behavior under growth. While finite element simulations seem to be capable of predicting the behavior of growing continua, they often cannot naturally capture the instabilities caused by growth. The accepted strategy to provoke growth-induced instabilities is therefore to perturb the solution of the problem, which indeed results in geometric instabilities in the form of wrinkles and folds. However, this strategy is intrinsically subjective as the user is prescribing the perturbations and the simulations are often highly perturbation-dependent. We propose a different strategy that is inherently suitable for this problem, namely eigenvalue analysis. The main advantages of eigenvalue analysis are that first, no arbitrary, artificial perturbations are needed and second, it is, in general, independent of the time step size. Therefore, the solution obtained by this methodology is not B C. Linder subjective and thus, is generic and reproducible. Equipped with eigenvalue analysis, we are able to compute precisely the critical growth to initiate instabilities. Furthermore, this strategy allows us to compare different finite elements for this family of problems. Our results demonstrate that linear elements perform strikingly poorly, as compared to quadratic elements.

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On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids

Cited by 31 publications

References 42 publications

On transformations and shape functions for enhanced assumed strain elements

On transformations and shape functions for enhanced assumed strain elements

Mixed Arlequin method for multiscale poromechanics problems

Computational aspects of growth-induced instabilities through eigenvalue analysis

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